It is shown that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if λ0, λ1… which sum to 1, then almost surely all components in such graphs are small.Expand

This talk defines graph colouring, explains the probabilistic tools which are used to solve them, and why one would expect the type of tools used to be effective for solving the types of problems typically studied.Expand

The size of the giant component in the former case, and the structure of the graph formed by deleting that component is analyzed, which is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.Expand

The authors present a linear-time algorithm that satisfies F with probability 1-o(1) whenever c<(0.25)2/sup k//k and establish a threshold for 2-SAT: if k = 2 then F is satisfiable with probability1-o (1) Whenever c<1 and unsatisfiable with probabilities 1-O(1), whenever c>1.Expand

We show that the strong chromatic index of a graph with maximum degree�; is at most (2��)�2, for some�>0. This answers a question of Erdo�s and Ne�et�il.

It is shown that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞, which settles a problem of Erdos who conjectured, in 1976, that A( G) = o(d2) as d →∞.Expand

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\\mathbfEM_n$ to within O(1) and prove exponential tail bounds for… Expand

It is shown that there exist constants α = 4.311… and β = 1.953 such that E(H<inf>n</inf></i) = α<i>ln n</i> − β(i) ln n + O(1), and thatVar(H) = <i>O</i>(1), which indicates the height of a random binary search tree on H(n) nodes.Expand